Let there be \(t\)  sets each with size at most \(a\) , and each element of the size \(n\)  universe is covered by at least \(b\)  of the sets, then there exists subset of \(O(\frac{t}{b}\log n)\)  size that covers the universe, and can be found with high probability using a Monte Carlo algorithm that runs in \(\tilde{O}(\frac{t}{b})\)  time.

[Sketch]
The condition shows the set cover LP has a feasible solution where every indicator variable for each set has value \(\frac{1}{b}\) . The standard randomized rounding algorithm of picking each set with probability equals \(\frac{1}{b}\)  independently, for \(\Theta (\log n)\)  rounds, would cover the universe with high probability [1]}. It can be simulated through independently sample sets of size \(\frac{t}{b}\)  for \(\Theta (\log n)\)  rounds instead, which can be done in \(\tilde{O}(\frac{t}{b})\)  time.
